Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium

We find solutions $E:\mathrm{\Omega }\to {\mathbb{R}}^3$ of the problem \begin{eqnarray*} \left{ \begin{aligned} &\nabla\times(\mu(x)^{-1}\nabla\times E) - \omega^2\epsilon(x) E = \partial_E F(x,E) &&\quad $\text{in }$\Omega\%\newline &\nu\times E = 0 &&\quad $\text{on }$\partial\Omega \end{aligned} \right. \end{eqnarray*} on a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with exterior normal $\nu :\mathrm{\partial }\mathrm{\Omega }\to {\mathbb{R}}^3$. Here $\mathrm{\nabla }\times$ denotes the curl operator in ${\mathbb{R}}^3$. The equation describes the propagation of the time-harmonic electric field $\mathrm{\Re }\{E(x)e^{i\omega t}\}$ in an anisotropic material with a magnetic permeability tensor $\mu (x)\in {\mathbb{R}}^{3\times 3}$ and a permittivity tensor $ϵ(x)\in {\mathbb{R}}^{3\times 3}$. The boundary conditions are those for $\mathrm{\Omega }$ surrounded by a perfect conductor. It is required that $\mu (x)$ and $ϵ(x)$ are symmetric and positive definite uniformly for $x\in \mathrm{\Omega }$, and that $\mu,\epsilon\in L^{\infty}(\Omega,$\mathbb{R}$^{3\times 3})$\mathrm{.}Thenonlinearity$F:\Omega\times$\mathbb{R}$^3\to$\mathbb{R}$$issuperquadraticandsubcriticalin$E$,themodelnonlinearitybeingofKerr-type:$F(x,E)=|\Gamma(x)E|^p$ for some $2<p<6$ with $\mathrm{\Gamma }(x)\in GL(3)$ invertible for every $x\in \mathrm{\Omega }$ and $\mathrm{\Gamma },{\mathrm{\Gamma }}^{-1}\in L^{\mathrm{\infty }}(\mathrm{\Omega },{\mathbb{R}}^{3\times 3})$. We prove the existence of a ground state solution and of bound states if $F$ is even in $E$. Moreover if the material is uniaxial we find two types of solutions with cylindrical symmetries.